Character sums and the series L(1,χ) with applications to real quadratic fields

In this article, let k = 0 or 1 (mod 4) be a fundamental discriminant, and let x(n) be the real even primitive character modulo k. The series can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0 ≤ j ≤ k - 1, and let Then. In section 2, Theorems 2.1 and 2.2 reveal a surprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that T(v, j, χ) · M < 0 for integer if and |M | ≥ 3/2. In section 3, we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4, we will attempt to make a connection between two conjectures on real quadratic fields and the sign of T(0, 20, χ).